Filetype PDF | Posted on 20 Jan 2023 | 2 years ago
Authentication
digital resources
136x Filetype PDF File size 0.49 MB Source: prog.lmu.edu.ng
BOUNDARY LAYER CONCEPT IN THE STUDY OF FLUID FLOW
When fluids flow over surfaces, the molecules near the surface are brought to rest
due to the viscosity of the fluid. The adjacent layers are also slow down, but to a lower
and lower extent. This slowing down is found limited to a thin layer near the surface. The
fluid beyond this layer is not affected by the presence of the surface. The fluid layer near
the surface in which there is a general slowing down is defined as boundary layer. The
velocity of flow in this layer increases from zero at the surface to free stream velocity at
the edge of the boundary layer.
When a real fluid flow past a solid body or a solid wall, the fluid particles adhere
to the boundary and condition of no slip occurs. This means that the velocity of fluid
close to the boundary will be same as that of the boundary. If the boundary is stationary,
the velocity of fluid at the boundary will be zero. The theory dealing with boundary layer
flows is called boundary layer theory.
According to the B.L. theory, the flow of fluid in the neighbourhood of the solid
boundary may be divided into two regions as shown below
Description of the Boundary Layer
The simplest boundary layer to study is that formed in the flow along one side of
a thin, smooth, flat plate parallel to the direction of the oncoming fluid. No other solid
surface is near, and the pressure of the fluid is uniform. If the fluid were inviscid no
velocity gradient would, in this instance, arise. The velocity gradients in a real fluid are
therefore entirely due to viscous action near the surface.
The fluid, originally having velocity U in the direction of plate, is retarded in
the neighbourhood of the surface, and the boundary layer begins at the leading edge of
the plate. As more and more of the fluid is slowed down, the thickness of the layer
increases. The fluid in contact with the plate surface has zero velocity, ‘no slip’ and a
velocity gradient exists between the fluid in the free stream and the plate surface.
The flow in the first part of the boundary layer (close to the leading edge of the
plate) is entirely laminar. With increasing thickness, however, the laminar layer becomes
unstable, and the motion within it becomes disturbed. The irregularities of the flow
develop into turbulence, and the thickness of the layer increases more rapidly. The
changes from laminar to turbulent flow take place over a short length known as the
transition region.
Graph of velocity u against distance y from surface at point X
Reynolds’ Number Concept
If the Reynolds number locally were based on the distance from the leading edge
of the plate, then it will be appreciated that, initially, the value is low, so that the fluid
flow close to the wall may be categorized as laminar. However, as the distance from the
leading edge increases, so does the Reynolds number until a point is reached where the
flow regime becomes turbulent.
For smooth, polished plates the transition may be delayed until Re equals 500000.
However, for rough plates or for turbulent approach flows transition may occur at much
lower values. Again, the transition does not occur in practice at one well-defined point
but, rather, a transition zone is established between the two flow regimes.
The figure above also depicts the distribution of shear stress along the plate in the
flow direction. At the leading edge, the velocity gradient is large, resulting in a high shear
stress. However, as the laminar region progresses, so the velocity gradient and shear
stress decrease with thickening of the boundary layer. Following transition the velocity
gradient again increases and the shear stress rises.
Theoretically, for an infinite plate, the boundary layer goes on thickening
indefinitely. However, in practice, the growth is curtailed by other surfaces in the
vicinity.
Factors affecting transition from Laminar to Turbulent flow Regimes
As mentioned earlier, the transition from laminar to turbulent boundary layer
condition may be considered as Reynolds number dependent, Re Usx x and a
x
5
figure of 5 x 10 is often quoted.
However, this figure may be considerably reduced if the surface is rough. For Re
5 5
<10 , the laminar layer is stable; however, at Re near 2 x 10 it is difficult to prevent
transition.
The presence of a pressure gradient dpdx can also be a major factor. Generally, if
dpdxis positive, then transition Reynolds number is reduced, a negative dpdx
increasing transition Reynolds number.
Boundary Layer thickness ( )
The velocity within the boundary layer increases from zero at the boundary
surface to the velocity of the main stream asymptotically. Therefore the thickness of the
boundary layer is arbitrarily defined as that distance from the boundary in which the
velocity reaches 99 per cent of the velocity of the velocity of the free stream
(u 0.99U). It is denoted by the symbol . This definition however gives an
approximate value of the boundary layer thickness and hence is generally termed as
nominal thickness of the boundary layer.
The boundary layer thickness for greater accuracy is defined as in terms of certain
mathematical expression which are the measure of the boundary layer on the flow. The
commonly adopted definitions of the boundary layer thickness are:
1. Displacement thickens ( *)
2. Momentum thickness ( )
3. Energy thickness (c)
- Displacement thickness ( *)
The displacement thickness can be defined as the distance measured
perpendicular to the boundary by which the main/free stream is displaced on account of
formation boundary layer.
Or
It is an additional “Wall thickness” that would have to be added to compensate for
the reduction in flow rate on account of boundary layer formation”.
Displacement thickness
Let fluid of density flow past a stationary plate with velocity U as shown
above. Consider an elementary strip of thickness dry at a distance y from the plate.
Assumed unit width, the mass flow per second through the elementary strip
udy i
Mass of flow per second through the elementary strip (unit width) if the plate
were not there
udy ii
Reduce the mass flow rate through the elementary strip
udyudy
uu dy
Total momentum of mass flow rate due to introduction of plate
U u dy iii
0
(If the fluid is incompressible)
Let the plate is displaced by a distance * and velocity of flow for the distance
* is equal to the main/free stream velocity (i.e. U). Then, loss of the mass of the fluid/sec.
*.
flowing through the distance
U* iv
Equating eqns. (iii) and (iv) we get
U* U u dy
0
or
* 1 u dy
0 U
Momentum Thickness ( )
This is defined as the distance which the total loss of momentum per second be
equal to if it were passing a stationary plate. It is denoted by .
It may also be defined as the distance, measured perpendicular to the boundary of
the solid body by which the boundary should be displaced to compensate for reduction in
momentum of the flowing fluid on account of boundary layer formation.
Refer to diagram of displacement thickness above,
no reviews yet
Please Login to review.
